Calculus

From Exampleproblems

I recommend this book: A Course of Modern Analysis by Whittaker and Watson. You may also find this book at Google Books. This book is a hundred years old and is considered the classic calculus book.

Derivatives

Definition of Derivative

$f'(x)=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$ , provided the limit exists.

For the following problems, find the derivative using the definition of the derivative.

solution $f(x)=x\,$

solution $f(x)=2x^3\,$

solution $f(x)=\sqrt{x}\,$

solution $f(x)=\frac{1}{x}\,$

solution $f(x)=\sin x\,$

solution $h(x)=f(x)+g(x)\,$

Power Rule

$\frac{d}{dx}(x^n)=nx^{n-1}\,$

For the following problems, compute the derivative of $y=f(x)\,$ with respect to $x\,$

solution $f(x)=150\pi\,$

solution $f(x)=7x^4\,$

solution $f(x)=27x^2+\frac{12}{\pi}x+e^{e^{\pi}}\,$

solution $f(x)=8x^5+4x^4+6x^3+x+7\,$

solution $f(x)=\frac{1}{\sqrt [4] {x}}\,$

solution $f(x)=7x^{-4}+12x^{-2}-x^{\frac{1}{2}}+6x+\pi x^{\frac{3}{5}}+x^{-\frac{3}{4}}\,$

solution Derive the power rule for positive integer powers from the definition of the derivative (Hint: Use the Binomial Expansion)

Product Rule

$\frac{d}{dx}(ab)=ab'+a'b\,$

For the following problems, compute the derivative of $y=f(x)\,$ with respect to $x\,$

solution $f(x)=x\sin x\,$

solution $f(x)=x^3\cos x\,$

solution $f(x)=\tan x\sec x\,$

solution $f(x)=2x\sin (2x)\,$

solution $f(x)=(x)(x+7)(x-12)\,$

solution Derive the Product Rule using the definition of the derivative

Quotient Rule

$\left(\frac{a}{b}\right)'=\frac{ba'-ab'}{b^2}\,$

For the following problems, compute the derivative of $y=f(x)\,$ with respect to x

solution $f(x)=\frac{x}{x+1}\,$

solution $f(x)=\frac{x^2}{\sin x}\,$

solution $f(x)=\frac{\sin^2x}{x^3}\,$

solution $f(x)=\frac{x\sin x}{e^x}\,$

solution $f(x)=\frac{x+7}{(x-6)(x+2)}\,$

solution Derive the Quotient Rule formula. (Hint: Use the Product Rule).

Generalized Power Rule

$\frac{d}{dx}(f(x))^n=n(f(x))^{n-1}f'(x)\,$

For the following problems, compute the derivative of $y=f(x)\,$ with respect to x

solution $f(x)=(x^3+7x)^3\,$

solution $f(x)=\sin^4(x)\,$

solution $f(x)=\left(\ln x\right)^{-4}\,$

solution $f(x)=\tan^2x+8(x^2+4x+3)^9+\sec^3x\,$

solution $f(x)=\left(\frac{5x}{7x+9}\right)^3\,$

Chain Rule

$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)\,$

For the following problems, compute the derivative of $y=f(x)\,$ with respect to x

solution $f(x)=\ln (7x^2e^x\sin x)\,$

solution $f(x)=\sin^2(7x+5) + \cos^2(7x+5)\,$

solution $f(x)=6e^{3x}\tan(5x)\,$

solution $f(x)=\ln (\sin (e^x))\,$

solution $f(x)=\sin (\cos (\tan x))\,$

solution $f(x)=e^{x^2}\sin (14x)-\cos (e^x)\,$

solution $f(x)=\frac{\frac{\sin (5x)}{(x^2+1)^2}}{\cos^3(3x)-1}\,$

solution $\left ( f(g(x)) \right )' = f'(g(x))g'(x)$

solution $\sqrt[]{x+\sqrt[]{x+\sqrt[]{x}}}$

solution $f(x)=x^2\sqrt{9-x^2}\,$

Implicit Differentiation

For the following problems, compute the derivative of $y=f(x)\,$ with respect to x

solution $\sin (xy)=x\,$

solution $x+xy+x^2+xy^2=0\,$

solution $y=x(y+1)\,$

solution $yx=x^y\,$ where $y\,$ is a function of $x\,$

Logarithmic Differentiation

For the following problems, compute the derivative of $y=f(x)\,$ with respect to x

solution $f(x)=4^{\sin x}\,$

solution $f(x)=x^x\,$

solution $f(x)=x^{x^{{\cdot}^{{\cdot}^{\cdot}}}} \,$

solution $f(x)=g(x)^{h(x)}\,$ for any functions $g(x)\,$ and $h(x)\,$ where $g(x) \ne 0\,$

Second Fundamental Theorem of Calculus

$\frac{d}{dx}\int_{a}^{x}f(t)\,dt=f(x)\,$

For the following problems, compute the derivative of $y=f(x)\,$ with respect to x

solution $f(x)=\int_{0}^{x}e^t\,dt\,$

solution $f(x)=\int_{5}^{x}27t^t\sin (t-1)\,dt\,$

solution $f(x)=\int_{4}^{x^2}\sin (e^t)\,dt\,$

solution $f(x)=\int_{x^2}^{3x^4}\cos (t)\,dt\,$

solution Give a proof of the theorem

Applications of Derivatives

Slope of the Tangent Line

solution Find the slope of the tangent line to the graph $f (x) = 6 x$ when $x = 3$ .

solution Find the slope of the tangent line to the graph $f (x) = sin 2 x$ when $x = π$ .

solution Find the slope of the tangent line to the graph $f (x) = cos x + x 5$ when $x=\frac{\pi}{2}$ .

solution Find the equation of the tangent line to the graph $f (x) = x e x + x + 5$ when $x = 0$ .

solution Find the slope of the tangent line to the graph $f (x) = x 2$ when $x = 0,1,2,3,4,5,6$ .

solution Find the slope of the tangent line to the graph $x 2 + y 2 = 9$ at the point $(0, - 3)$ .

solution Find the equation of the tangent line to the graph $x y = y 2 x 2 + y + x + 2$ at the point $(0, - 2)$ .

Extrema (Maxima and Minima)

solution Find the absolute minimum and maximum on $[ - 1,5]$ of the function $f (x) = (1 - x) e x$ .

solution Find the absolute minimum and maximum on $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ of the function $f (x) = sin(x 2)$ .

solution Find all local minima and maxima of the function $f (x) = x 2$ .

solution Find all local minima and maxima of the function $f(x)=\frac{x^2-1}{x}$ .

solution If a farmer wants to put up a fence along a river, so that only 3 sides need to be fenced in, what is the largest area he can fence with 100 feet of fence?

solution If a fence is to be made with three pens, the three connected side-by-side, find the dimensions which give the largest total area if 200 feet of fence are to be used.

solution Find the local minima and maxima of the function $f(x)=\sqrt {x}$ .

Related Rates

solution A clock face has a 12 inch diameter, a 5.5-inch second hand, a 5 inch minute hand and a 3 inch hour hand. When it is exactly 3:30, calculate the rate at which the distance between the tip of any one of these hands and the 9 o'clock position is changing.

solution A spherical container of $r$ meters is being filled with a liquid at a rate of $\rho\,{\rm m}^3/{\rm min}$ . At what rate is the height of the liquid in the container changing with respect to time?

Projectile Motion

Integrals

Riemann Sums

Integration by Substitution

solution $\int \frac{2x}{x^2+1}\,dx\,$

solution $\int x^2\sin x^3\,dx\,$

solution $\int \cot x\,dx\,$

solution $\int \tan x\sec^2x\,dx\,$

solution $\int \frac{\ln x^2}{x}\,dx\,$

solution $\int_{0}^{3} \frac{2x+1}{x^2+x+7}\,dx\,$

solution $\int_{1}^{e^{\pi}} \frac{\sin (\ln x)}{x}\,dx\,$

solution $\int \frac{x}{\sqrt{4+x^2}}\,dx\,$

solution $\int \frac{x}{1-x^2}\,dx\,$

Integration by Parts

$\int u\,dv=uv-\int v\,du\,$

solution $\int \ln x\,dx\,$

solution $\int x\sin(x)\,dx \,$

solution $\int \arctan(2x)\,dx \,$

solution $\int e^x\sin x\,dx\,$

solution $\int x^2e^x\,dx\,$

solution $\int (x^3+1)\cos x\,dx\,$

solution $\int x^4\sin x\,dx\,$

solution $\int \frac{\ln x}{x}\,dx\,$

Trigonometric Integrals

solution $\int \sin^2(x)\,dx\,$

solution $\int \tan^2(x)\,dx \,$

solution $\int \sin^5x\cos^5x\,dx\,$

solution $\int \sin^2x\cos^3x\,dx\,$

solution $\int \sin^2x\cos^2x\,dx\,$

solution $\int \tan^2x\sec^4x\,dx\,$

solution $\int \tan^3x\sec^3x\,dx\,$

Trigonometric Substitution

solution $\int\frac{x\,dx}{\sqrt{3-2x-x^2}}$

solution $\int \arcsec x\,dx \,$

solution $\int \frac{1}{\sqrt{4x-x^2}}\,dx\,$

solution $\int x\arcsin x\,dx\,$

solution $\int \frac{x}{1-x^2}\,dx\,$

solution $\int \frac{1}{(x^2+1)^{\frac{3}{2}}}\,dx\,$

solution $\int \frac{\sqrt{x^2-1}}{x}\,dx\,$

solution $\int \frac{x}{\sqrt{4+x^2}}\,dx\,$

Partial Fractions

solution $\int\frac{1}{x^2-1}\,dx\,$

solution $\int\frac{1}{(x+1)(x+2)(x+3)}\,dx\,$

solution $\int\frac{x!}{(x+n)!}\,dx\,$

solution $\int \frac{x}{1-x^2}\,dx\,$

solution $\int\frac{dx}{\sin^2(x)-\cos^2(x)}$

Special Functions

solution A ball is thrown up into the air from the ground. How high will it go?

solution Let $f\,$ be a continuous function for $x \ge a\,$ .
Show that $\int_a^x \int_a^s f(y)\,dy\,ds = \int_a^x f(y)(x-y)\,dy\,$

solution Evaluate $\int e^{x^2}\,dx\,$

solution $\int_0^\infty 3^{-4z^2}dz\,$

solution $\int_0^\infty x^m e^{-ax^n} dx\,$

solution $\int_0^\infty x^4 e^{-x^3} \,dx\,$

solution $\int_0^\infty e^{-pt} \sqrt{t}\,dt\,$

Applications of Integration

Area Under the Curve

solution Find the area under the curve $f (x) = x 2$ on the interval $[ - 1,1]$ .

solution Find the total area between the curve $f (x) = cos x$ and the x-axis on the interval $[0,2π]$ .

solution Find the area under the curve $f (x) = x 3 + 4 x 2 - 7 x + 8$ on the interval $[0,1]$ .

solution Using calculus, find the formula for the area of a rectangle.

solution Derive the formula for the area of a circle with arbitrary radius r.

Volume

Disc Method

The disc method is a special case of the method of cross-sectional areas to find volumes, using a circle as the cross-section.

To find the volume of a solid of revolution, using the disc method, use one of the two formulas below. $R (x)$ is the radius of the cross-sectional circle at any point.

If you have a horizontal axis of revolution

$V=\pi\int_{a}^{b}[R(x)]^2\,dx$

If you have a vertical axis of revolution

$V=\pi\int_{c}^{d}[R(y)]^2\,dy$

solution Find the volume of the solid generated by revolving the line $y = x$ around the x-axis, where $0\le x\le 4$ .

solution Find the volume of the solid generated by revolving the region bounded by $y = x 2$ and $y = 4 x - x 2$ around the x-axis.

solution Find the volume of the solid generated by revolving the region bounded by $y = x 2$ and $y = x 3$ around the x-axis.

solution Find the volume of the solid generated by revolving the region bounded by $y = x 2$ and $y = x 3$ around the y-axis.

solution Find the volume of the solid generated by revolving the region bounded by $y=\sqrt{x}$ , the x-axis and the line $x = 4$ around the x-axis.

Shell Method

Cross-sectional Areas

solution Find the volume, on the interval $0\le x\le 3$ , of a 3-D object whose cross-section at any given point is a square with side length $x 2 - 9 x$ .

solution Find the volume, on the interval $0 < x < 2π$ , of a 3-D object whose cross-section at any given point is an equilateral triangle with side length $\sin \frac{x}{2}$ .

solution Find the volume of a cylinder with radius 3 and height 10.

Arc Length

$L=\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^2} \,dx = \int_p^q\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2} \,dt$

solution Calculate the arc length of the curve $y = x 2$ from $x = 0$ to $x = 4$ .

solution Determine the arc length of the curve given by x = t cos t , y = t sin t from t=0

solution Calculate the arc length of y=cosh x from x=0 to x

Mean Value Theorem

$f ' (c) = \frac{f(b) - f(a)}{b - a}.$

solution Find the average value of the function $f (x) = e 2 x$ on the interval $[0,4]$ .

solution Find the average value of the function $2 s e c 2 x$ on the interval $[0,\frac{\pi}{4}]$ .

solution Find the average speed of a car, starting at time 0, if it drives for 5 hours and its speed at time t (in hours) is given by $s (t) = 5 t 2 + 7 t + e t$ .

solution Find the average value of the function $sin 6 t c o s 3 t$ on the interval $[0,\frac{\pi}{2}]$ .

solution Deduce the Mean Value Theorem from Rolle's Theorem.

Series of Real Numbers

Sequences

nth Term Test

If the series $\sum_{n=1}^{\infty}a_n$ converges, then $\lim_{n\rightarrow \infty}a_n=0$ .

Note: This leads to a test for divergence for those series whose terms do not go to $0$ but it does not tell us if any series converges.

solution Discuss the convergence or divergence of the series with terms ${ - 1,1, - 1,1, - 1,1,...}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{1}{n}$ and $\sum_{n=1}^{\infty}\frac{1}{n^2}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=0}^{\infty}\sin n$ .

Telescopic Series

If ${a k}$ is a convergent real sequence, then $\sum_{k=1}^{\infty}(a_k-a_{k+1})=a_1-\lim_{k\rightarrow \infty}a_k$ .

solution Discuss the convergence or divergence of the series $\sum_{k=1}^{\infty}\left(\frac{1}{k+7}-\frac{1}{k+8}\right)$ .

solution Discuss the convergence or divergence of the series $\sum_{k=0}^{\infty}\frac{2}{(k+1)(k+3)}$ .

solution Discuss the convergence or divergence of the series $\sum_{k=2}^{\infty}\ln \left(\frac{k(k+2)}{(k+1)^2}\right)$ .

solution Discuss the convergence or divergence of the series $\sum_{k=4}^{\infty}\left(\frac{1}{k}-\frac{1}{k+2}\right)$ .

solution Discuss the convergence or divergence of the series $\sum_{k=0}^{\infty}\left(e^{-k}-e^{-(k+1)}\right)$ .

solution Discuss the convergence or divergence of the series $\sum_{k=4}^{\infty}\left(e^{-k+3}-e^{-k+1}\right)$ .

Geometric Series

The series $\sum_{n=0}^{\infty}r^n$ converges if $- 1 < r < 1$ and, moreover, it converges to $\frac{1}{1-r}$ . For any other value of r, the series diverges. More generally, the finite series, $\sum_{n=a}^{b}r^n=\frac{r^a-r^{b+1}}{1-r}$ for any value of $r$ .

solution Discuss the convergence or divergence of the series $\sum_{n=0}^{\infty}\left (\frac{1}{4}\right )^n$ .

solution Discuss the convergence or divergence of the series $\sum_{n=3}^{\infty}\left (\frac{2}{3}\right )^n$ .

solution Discuss the convergence or divergence of the series $\sum_{n=0}^{\infty}1.5^n$ .

solution Discuss the convergence or divergence of the series $\sum_{n=0}^{\infty}\left (\frac{1}{6}\right )^{n+2}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=10}^{\infty}\left (\frac{3}{5}\right )^n$ .

solution Discuss the convergence or divergence of the series $\sum_{n=7}^{\infty}\left (-\frac{1}{2}\right )^n$ .

solution Discuss the convergence or divergence of the series $\sum_{n=7}^{\infty}\left[\left (-\frac{4}{7}\right )^{n+3}+\left (\frac{1}{3}\right )^{n-2}\right]\,$ .

solution Discuss the convergence or divergence of the series $\sum_{n=0}^{\infty}\frac{3^{n+1}}{7^n}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=0}^{\infty}\left (\frac{1}{3}\right )^{2n}$ .

Integral Test

If the function $f$ is positive, continuous, and decreasing for $x\ge 1$ , then

$\sum_{n=1}^{\infty}f(n)\,$ and $\int_{1}^{\infty}f(x)\,dx\,$

converge together or diverge together.

Notice that if $f$ were negative, continuous, and increasing this is also true since such a function would simply be the negative of some function which is positive, continuous, and decreasing and multiplying by -1 will not change the convergence of a series.

solution Discuss the convergence of $\sum_{n=1}^{\infty}\frac{1}{n+1}$ .

solution Discuss the convergence of $\sum_{n=0}^{\infty}e^{-3n}$ .

solution Discuss the convergence of $\sum_{n=1}^{\infty}\frac{1}{n^5}$ .

solution Discuss the convergence of $\sum_{n=1}^{\infty}\frac{n}{e^n}$ .

solution Explain why the integral test is or is not applicable to $\sum_{n=1}^{\infty}n^2$ .

solution Explain why the integral test is or is not applicable to $\sum_{n=1}^{\infty}-\frac{1}{n^2}$ .

solution Explain why the integral test is or is not applicable to $\sum_{n=1}^{\infty}\frac{|\sin n|+1}{\ln (n+1)}$ .

solution Discuss the convergence of $\sum_{n=1}^{\infty}\frac{1}{n^p}$ .

solution Discuss the convergence of $\sum_{n=1}^{\infty}\frac{1}{n^{\frac{7}{3}}}$ .

solution Discuss the convergence of $\sum_{n=1}^{\infty}\frac{1}{\sqrt [3] {n}}$ .

Comparison of Series

Direct Comparison

If $0<a_n\le b_n$ for all

n

If $\sum_{n=1}^{\infty}b_n$ converges, then $\sum_{n=1}^{\infty}a_n$ also converges.

2. If $\sum_{n=1}^{\infty}a_n$ diverges, then $\sum_{n=1}^{\infty}b_n$ also diverges.

Limit Comparison

Suppose

a n > 0

,

b n > 0

and $\lim_{n\rightarrow \infty}\frac{a_n}{b_n}=L$ where

L

is finite and positive.

Then $\sum_{n=1}^{\infty}a_n$ and $\sum_{n=1}^{\infty}b_n$ either both converge or both diverge.

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{1}{n^3+4}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{\ln n}{n-3}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{2n^2}{4n^4+8n^3+3}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{1}{n\sqrt{n^2+1}}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{n+4}{(n+2)(n+1)}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\sin \frac{1}{n}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{n+1}{n\times 3^{n+1}}$ .

Dirichlet's Test

Let $a_k, b_k\in \mathbb{R}$ for $k\in \mathbb{N}$ .

If the sequence of partial sums $s_n=\sum_{k=1}^{n}a_k$ is bounded and $b_k\downarrow 0$ as $k\rightarrow \infty$ , then $\sum_{k=1}^{\infty}a_kb_k$ converges.

solution Discuss the convergence or divergence of the series $\sum_{k=1}^{\infty}\frac{\sin \frac{k\pi}{2}}{k}$ .

solution Discuss the convergence or divergence of the series $\sum_{k=2}^{\infty}\frac{\sin \frac{k\pi}{3}}{\ln k}$ .

solution Discuss the convergence or divergence of the series $\sum_{k=7}^{\infty}\left(-\frac{1}{2}\right)^k$ .

solution Discuss the convergence or divergence of the series $\sum_{k=1}^{\infty}\frac{1}{k2^k}$ .

solution Discuss the convergence or divergence of the series $\sum_{k=3}^{\infty}\frac{1}{\ln (\ln k)}\cos \left(\frac{k\pi}{3}\right)$ .

solution Discuss the convergence or divergence of the series $\sum_{k=1}^{\infty}a_k\frac{1}{k^3}$ where $a k = {7,4,6,3, - 10, - 10,7,4,6,3, - 10, - 10,...}$ .

solution Discuss the convergence or divergence of the series $\sum_{k=1}^{\infty}\sin \frac{2k\pi}{n}\frac{1}{\ln (\ln (\ln k))}$ for any integer $n\ge 1$ .

solution Discuss the convergence or divergence of the series $\sum_{k=3}^{\infty}\frac{\ln (\ln k)}{\ln k}a_k$ where ${a k} = {1, - 1,1,2, - 3,1,2,4, - 7,1, - 1,1,2, - 3,1,2,4, - 7,...}$ .

Alternating Series

If $a n$ , then the alternating series

$\sum_{n=1}^{\infty}(-1)^na_n\,$ and $\sum_{n=1}^{\infty}(-1)^{n+1}a_n\,$

converge if the absolute value of the terms decreases and goes to $0$ .

solution Discuss the convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\,$ .

solution Discuss the convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}n^2}{n^2+1}\,$ .

solution Discuss the convergence of the series $\sum_{n=1}^{\infty}\cos (n\pi)\,$ .

solution Discuss the convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}}\,$ .

solution Discuss the convergence of the series $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}n^2}{e^n}\,$ .

solution Discuss the convergence of $\sum_{n=2}^{\infty}\frac{(-1)^n}{\ln n}$ including whether the sum converges absolutely or conditionally.

solution Discuss the convergence of $\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^2+3n+4}$ including whether the sum converges absolutely or conditionally.

solution Discuss the convergence of $\sum_{n=1}^{\infty}\frac{(-1)^n}{n!}$ including whether the sum converges absolutely or conditionally.

Ratio Test

For the infinite series $\sum_{n=1}^{\infty}a_n$ ,

1. If $0\le \lim_{n\rightarrow \infty}\left|\frac{a_{n+1}}{a_n}\right|<1\,$ , then the series converges absolutely.

2. If $\lim_{n\rightarrow \infty}\left|\frac{a_{n+1}}{a_n}\right|>1\,$ , then the series diverges.

3. If $\lim_{n\rightarrow \infty}\left|\frac{a_{n+1}}{a_n}\right|=1\,$ , then the test is inconclusive.

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{2^n}{(2n)!}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{n}{3^{n+1}}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{1}{n^3}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{1}{n}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{3^n}{n^2+2}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{n}{(n+2)!}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{n4^n}{n!}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{n!}{4^n}$ .

solution Discuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{n^ka^n}{n!}$ .

Root Test

Let $a_k\in\mathbb{R}$ and $r=\limsup_{k\rightarrow \infty}|a_k|^\frac{1}{k}$

1. If $r<1\,$ , then $\sum_{k=1}^{\infty}a_k$ converges absolutely.

2. If $r>1\,$ , then $\sum_{k=1}^{\infty}a_k$ diverges.

3. If $r=1\,$ , this test is inconclusive.

solutionDiscuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\left(\frac{n}{2n+1}\right)^n$ .

solutionDiscuss the convergence or divergence of the series $\sum_{n=1}^{\infty}e^{-n}$ .

solutionDiscuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{n}{3^n}$ .

solutionDiscuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\left(\frac{2n}{100}\right)^n$ .

solutionDiscuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{(n!)^n}{(n^n)^2}$ .

solutionDiscuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\left(\frac{7n}{12n-6}\right)^{2n}$ .

solutionDiscuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\left(\frac{15n-6}{4n+2}\right)^{7n}$ .

solutionDiscuss the convergence or divergence of the series $\sum_{n=1}^{\infty}\frac{n!}{(n+1)^n}\left(\frac{23}{50}\right)^n$ .

Cauchy Condensation Test

The series $\sum_{k=1}^{\infty}a_k$ and $\sum_{k=1}^{\infty}2^ka_{2^k}$ converge or diverge together.

solutionDiscuss the convergence or divergence of the series $\sum_{k=1}^{\infty}\frac{1}{k}$ .

solutionDiscuss the convergence or divergence of the series $\sum_{k=1}^{\infty}\frac{1}{\ln k}$ .

solutionDiscuss the convergence or divergence of the series $\sum_{k=1}^{\infty}\frac{1}{k^2}$ .

solutionDiscuss the convergence or divergence of the series $\sum_{k=1}^{\infty}\frac{1}{\sqrt{2^k}}$ .

solutionDiscuss the convergence or divergence of the series $\sum_{k=1}^{\infty}\frac{1}{(2^k)^n}$ , where n is some real number.

Logarithmic Test

Raabe's Test

Series of Real Functions

solution Find the infinite series expansion of $\frac{1}{(1+x)^a}\,$

solution Investigate the convergence of this series: $\sum_{k=1}^\infty \frac{1}{k(k+1)}\,$

solution Investigate the convergence of this series: $3-2+\frac{4}{3}-\frac{8}{9}+...+3\left(-\frac{2}{3}\right)^k+... \,$

solution Find the upper limit of the sequence $\left\{x_n\right\}_{n=1}^\infty, x_n=(-1)^nn\,$

solution Find the upper limit of the sequence $\left\{x_n\right\}_{n=1}^\infty, x_n=(-1)^n\left(\frac{2n}{n+1}\right)\,$

solution Evaluate $\sum_{k=0}^n {n \choose k}^2\,$

solution Evaluate $\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{m^2\,n}{3^m\left(m\,3^n+n\,3^m\right)}$ .

solution Find the upper limit of the sequence $\left\{x_n\right\}_{n=1}^\infty, x_n=\frac{1}{n^2}\,$

solution Find the upper limit of the sequence $\left\{x_n\right\}_{n=1}^\infty, x_n=n\sin\left(\frac{n\pi}{2}\right)\,$

solution Evaluate $\sum_{n=0}^\infty \left(\frac{i}{3}\right)^n\,$

solution Determine the interval of convergence for the power series $\sum_{n=0}^\infty\frac{n(3x-4)^n}{\sqrt[3]{n^4}(2x)^{n-1}}.$